Stream Classification and Solubility of the Dispersion Equation for Piecewise Constant Vorticity

Abstract: This thesis concerns the water wave problem corresponding to a piecewise constant vorticity function. There are several results connected to this field. In [1] the authors prove the existence of small-amplitude capillary-gravity water waves in the setting of unidirectional waves, and present an explicit form of the dispersion equation in the case when the vorticity function has two jumps. A two-layer model with constant but different vorticities is studied in [2], while in [3], an analysis of the dispersion equation for a three-layer model is given. In this thesis we first classify all stream solutions to the problem specified above, and then use our classification to prove and analyze solubility of the dispersion equation for a vorticity function with one jump. We do not require streams to be unidirectional (that is, we allow underlying counter-currents and internal stagnation).